Errors and Statistics

Error Propagation

When dealing with uncertainties based on a large collection
of numbers, the manipulation of measured quantities and the error associated
with each quantity will contribute to the error in the final answer. The
following is an informal discussion describing how to make reasonable
approximations of errors associated with measuring physical quantities.

Error analysis is the study of error propagation within an
experiment and the full treatment is quite complicated and detailed. Up until
now, we have compared our results using percentage error and percentage
difference, but have largely ignored the error inherent in our measurements.
Let’s now discuss the topic of error propagation.

A. Uncertainty of an object’s mass. Say you use a
triple-beam balance to measure the mass of some object and you find that value
to be 156.28g. So is this the actual mass? We say this is the measured or
nominal mass value, but it is not the actual mass, for we are somewhat uncertain
about the measurement due to our instrument’s imperfections. Let’s find out how
uncertain.

Since the least count of the triple-beam balance is 0.1g, the
uncertainty of any measurement made with this instrument (no matter how
carefully the measurement is performed) is 0.05g, or 50% of the least count.
Therefore we would report the mass of this object to be 156.28g ± 0.05g. This
implies that we are certain that the mass of the object is somewhere
between 156.23g and 156.33g. Of course, this assumes the instrument is properly
zeroed and is in proper working order.

B. Uncertainty of a sphere’s volume. Say you use a
Vernier caliper to measure the diameter of solid sphere and your find that value
to be 4.22cm. Therefore the radius of that sphere is 2.11cm. Using the formula
we can calculate the volume of the sphere to be 39.349206cm3. Of
course we must take into account significant figures and round our answer to
39.35cm3. How certain are we that this measurement is correct? Let’s
calculate the uncertainty of our calculation using a "brute-force"
method.

We know from Section IV above that the uncertainty of any measurement taken
by a Vernier caliper is ± 0.01 cm. Therefore our original measurement of 2.11cm
really falls within the range of 2.11 ± 0.01cm. That is, we can say with a
great degree of certainty that the actual value of the sphere’s radius falls
between 2.10cm and 2.12cm. For example, if we plug in 2.10cm as the radius, we
find the lower-limit of the sphere’s volume to be 38.79cm3. This is
a difference of –0.56cm3 from the nominal measurement of 39.35cm3.
Plugging in 2.12cm we calculate the upper-limit of the sphere’s volume to be
39.91cm3, this time a difference of +0.56cm3 from the
nominal measurement. Here, both the lower and upper limits differ by 0.56cm3
from the nominal measurement; therefore the uncertainty of this
measurement is ± 0.56cm3. We are finally able to say that
the volume of the sphere is measured to be 39.35cm3 ± 0.56cm3.
Sometimes this is written 39.35 ± 0.56cm3.

Figure 14. The radius of a sphere is measured to be
2.11, and its volume is calculated to be 39.35cm3

With every measurement, however, there is an associated uncertainty.
Because a Vernier caliper was used here, the uncertainty in the radius
measurement is ± 0.01cm. Using the lower-limit of the sphere’s radius of
2.10cm, the lower-limit of the sphere’s volume is calculated to be 38.79cm3.
Using the radius’ upper-limit of 2.12cm, the upper-limit of the sphere’s
volume is calculated to be 30.91cm3. The difference between the lower
and upper limits and the nominal value is ± 0.56cm3, so we report
that the volume of the sphere is 39.35cm3 ± 0.56cm3.

While this method does not treat error propagation from a theoretical point
of view, the practical treatment here will suffice at present.

C. Uncertainty of the volume of a rectangular block. Let’s quickly
take a look at one more example of error propagation. Say we use a Vernier
caliper to measure the length, width, and height of a rectangular block as
1.37cm, 4.11cm, and 2.56cm, respectively. The nominal volume of the block is
therefore, 14.41cm3 ( ).

Of course, each of these measurements has an uncertainty of ± 0.01cm due to
imperfections in the caliper. As before we can calculate a lower-limit of the
block’s volume:

,

which is 0.19cm3 lower than the nominal value.

The upper-limit can also be calculated:

,

which is 0.20cm3 higher than the nominal value. We must
choose the highest deviation from the nominal to serve as our uncertainty, so we
would report that the volume of the block is 14.14cm3 ± 0.20cm3.

D. Reporting with error bars. When a quantity is
graphed, it is common for the uncertainty of that quantity to be represented by error
bars. Refer to the Excel tutorial on using error bars to learn how to
include this important piece of data into your laboratory reports. You should
know that if a quantity has the same uncertainty value for each
measurement, for example mass, you can enter a fixed error value as shown in
Figure 15. If each data point has a unique uncertainty value, for example
volume, you should store those in a column in the spreadsheet and enter the
cells of the custom values as shown in Figure 16.